what v hav 2 find the length of the chord which is from from the point of contact of the tangents with the curve for any type of curve.explain with parabolla and ellipse

Hello student,
please find the answer to your question below
suppose we have to find the length of chord of parabola y² =4ax
by the tangents from points (x1,y1)
so equation of chord of contact : yy1 = 2a(x+x1)
now find the point of intersection of chord of contact
and given parabola y² =4a(yy1 - 2ax1)/2a or y² -2yy1 +4ax1 =0
let point of intersectaion are (h1,k1) and (h2,k2)
k1 +k2 = 2y1 and k1k2 = 4ax1
now find k1 -k2 and we also know that
k1²= 4ah1 and k2²= 4ah²
so k1² -k2²= 4ah1- 4ah2 ( k1²-k2²)/4a= h1- h2
so length of chord = √[(h1-h2)² +(k1-k2)² ]
put the above value u will get length = (y1² -4ax1)(y1² +4a²)/a